The Aerodynamic boundary layer was first proposed by Ludwig Prandtl in a paper presented on August 12, 1904 at the Third International Congress of Mathematicians in Heidelberg, Germany. It simplifies the equations of fluid flow by dividing the flow field into two zones: one within the boundary layer, dominated by viscosity and creating most of the resistance "Resistance (fluids)") experienced by the boundary body; and another outside the boundary layer, where the viscosity can be neglected without significant effects on the solution. This allows a closed-form (mathematical) closed-form solution for the flow in both zones making significant simplifications of the full Navier-Stokes equations. The same hypothesis is applicable to other fluids (besides air) of moderate to low viscosity, such as water. In the case where there is a temperature difference between the surface and the fluid, it is observed that most of the heat transfer to and from a body takes place in the vicinity of the velocity boundary layer. Again, this allows the equations in the flow field outside the boundary layer to be simplified. The pressure distribution along the boundary layer in the direction normal to the surface (like an airfoil) remains relatively constant along the boundary layer, and is the same as at the surface itself.

The thickness of the velocity boundary layer is normally defined as the distance from the solid body to the point at which the viscous flow velocity is 99% of the free stream velocity (the surface velocity of an inviscid flow). The displacement thickness") is an alternative definition that states that the boundary layer represents a deficit in mass flow compared to non-viscous flow with wall slip. It is the distance the wall would have to be displaced in the non-viscous case to obtain the same total mass flux as in the viscous case. The no-slip condition requires that the flow velocity at the surface of a solid object be zero and that the fluid temperature be equal to the surface temperature. The flow velocity will then increase rapidly within the boundary layer, governed by the boundary layer equations, below.

The thickness of the thermal boundary layer is the distance from the body at which the temperature is 99% of the free stream temperature. The relationship between the two thicknesses is governed by the Prandtl number. If the Prandtl number is 1, the two boundary layers have the same thickness. If the Prandtl number is greater than 1, the thermal boundary layer is thinner than the velocity boundary layer. If the Prandtl number is less than 1, which is the case for air under standard conditions, the thermal boundary layer is thicker than the velocity boundary layer.

In high-performance designs, such as gliders and commercial aircraft, much attention is paid to controlling the behavior of the boundary layer to minimize drag. Two effects must be taken into account. First, the boundary layer increases the effective thickness of the body, through the displacement thickness"), thus increasing the pressure resistance. Second, the shear forces") on the wing surface create skin friction resistance.

At high Reynolds numbers, typical of large aircraft, it is desirable to have a laminar boundary layer. This results in lower skin friction due to the characteristic velocity profile of laminar flow. However, the boundary layer inevitably thickens and becomes less stable as the flow develops along the body, eventually becoming turbulent, a process known as the boundary layer transition. the boundary layer towards the stern by reshaping the airfoil or fuselage so that its thickest point is further aft and less thick. In this way, the speeds at the front are reduced and the same Reynolds number is achieved with a greater length.

At low Reynolds numbers, such as those seen in model airplanes, it is relatively easy to maintain laminar flow. This results in low surface friction, which is desirable. However, the same velocity profile that gives the laminar boundary layer its low skin friction also causes it to be greatly affected by adverse pressure gradients. When pressure begins to recover at the rear of the wing chord, the laminar boundary layer will tend to separate from the surface. Such flow separation causes a large increase in pressure resistance, since it greatly increases the effective size of the wing section. In these cases, it may be advantageous to deliberately cause boundary layer turbulence at a point prior to the laminar separation, using a turbulator. The more complete velocity profile of the turbulent boundary layer allows it to maintain the adverse pressure gradient without separating. Thus, although surface friction increases, overall resistance decreases. This is the principle behind the dimples in golf balls and the vortex generators in airplanes. Special wing sections have also been designed that adapt pressure recovery to reduce or even eliminate laminar separation. This represents an optimal compromise between the pressure resistance of flow separation and the surface friction of induced turbulence.

When using semi-models in wind tunnels, a peniche&action=edit&redlink=1 "Peniche (fluid dynamics) (not yet drafted)") is sometimes used to reduce or eliminate the boundary layer effect.

Contenido

La deducción de las ecuaciones de la capa límite fue uno de los avances más importantes en la dinámica de fluidos. Utilizando un análisis de orden de magnitud"), las conocidas ecuaciones de Navier-Stokes que gobiernan el flujo viscoso de fluidos pueden simplificarse en gran medida dentro de la capa límite. pueden simplificarse en gran medida dentro de la capa límite. En particular, el [característica de la Ecuaciones diferenciales parciales (EDP) se convierte en parabólica, en lugar de la forma elíptica de las ecuaciones de Navier-Stokes completas. Esto simplifica enormemente la solución de las ecuaciones. Al hacer la aproximación de la capa límite, el flujo se divide en una parte no viscosa (que es fácil de resolver por varios métodos) y la capa límite, que se rige por una EDP más fácil de resolver. Las ecuaciones de continuidad y de Navier-Stokes para un flujo incompresible bidimensional constante en coordenadas cartesianas vienen dadas por.

donde y son las componentes de la velocidad, es la densidad, es la presión, y es la viscosidad cinemática del fluido en un punto.

La aproximación establece que, para un número de Reynolds suficientemente alto, el flujo sobre una superficie puede dividirse en una región exterior de flujo no viscoso no afectado por la viscosidad (la mayor parte del flujo), y una región cercana a la superficie donde la viscosidad es importante (la capa límite). Sean y las velocidades de la corriente y transversal (normal a la pared) respectivamente dentro de la capa límite. Usando análisis de escala "Análisis de escala (matemáticas)"), se puede demostrar que las ecuaciones de movimiento anteriores se reducen dentro de la capa límite a ser.

y si el fluido es incompresible (como los líquidos en condiciones normales):.

El análisis del orden de magnitud supone que la escala de longitud en el sentido de la corriente es significativamente mayor que la escala de longitud transversal dentro de la capa límite. De ello se deduce que las variaciones de las propiedades en la dirección de la corriente son generalmente mucho menores que las de la dirección normal de la pared. Si se aplica esto a la ecuación de continuidad, se observa que , la velocidad normal de la pared, es pequeña comparada con , la velocidad en el sentido de la corriente.

Como la presión estática es independiente de , entonces la presión en el borde de la capa límite es la presión en toda la capa límite en una posición dada en el sentido de la corriente. La presión externa puede obtenerse mediante una aplicación de la ecuación de Bernoulli. Sea la velocidad del fluido fuera de la capa límite, donde y son ambas paralelas. Esto da al sustituir por el siguiente resultado.

Para un flujo en el que la presión estática tampoco cambia en la dirección del flujo.

si permanece constante.

Por lo tanto, la ecuación de movimiento se simplifica y pasa a ser.

Estas aproximaciones se utilizan en una variedad de problemas prácticos de flujo de interés científico y de ingeniería. El análisis anterior es para cualquier laminar instantáneo o turbulento, pero se utiliza principalmente en estudios de flujo laminar ya que el flujo medio es también el flujo instantáneo porque no hay fluctuaciones de velocidad presentes. Esta ecuación simplificada es una EDP parabólica y puede resolverse utilizando una solución de similitud a menudo denominada capa límite de Blasius.

Prandtl transposition theorem

Prandtl observed that from any solution that satisfies the boundary layer equations, another solution, which also satisfies the boundary layer equations, can be constructed by writing [2]​.

where it is arbitrary. Since the solution is not unique from a mathematical point of view,[3]​ any of an infinite set of eigenfunctions can be added to the solution as demonstrated by Stewartson").[4]​ and Paul A. Libby").[5]​[6]​.

The treatment of turbulent boundary layers is much more difficult due to the time-dependent variation of flow properties. One of the most used techniques to treat turbulent flows is to apply the Reynolds decomposition. In it, the instantaneous properties of the flow are decomposed into an average and a fluctuating component, assuming that the average of the fluctuating component is always zero. Applying this technique to the boundary layer equations, the complete turbulent boundary layer equations are obtained, which are rare in the literature:

Using a similar order of magnitude analysis, the above equations can be reduced to leading order terms. Choosing length scales for changes in the transverse direction, and for changes in the direction of the current, with , the x-momentum equation simplifies to:.

This equation does not satisfy the no-slip condition on the wall. As Prandtl did for his boundary layer equations, a new smaller length scale must be used to allow the viscous term to become leading order in the momentum equation. Choosing y as the scale, the leading order moment equation for this "inner boundary layer" is given by:.

In the limit of infinite Reynolds number, it can be shown that the pressure gradient term has no effect in the inner region of the turbulent boundary layer. The new "internal length scale" is a viscous length scale, and is of order , being the speed scale of turbulent fluctuations, in this case a frictional speed.

Unlike the laminar boundary layer equations, the presence of two regimes governed by different sets of flow scales (i.e., the inner and outer scale) has made finding a universally similar solution for the turbulent boundary layer difficult and controversial. To find a similarity solution that spans both regions of the flow, it is necessary to asymptotically match the solutions of both regions of the flow. This analysis will result in the so-called Wall Law or Power Law Solutions.

Approaches similar to the previous analysis have also been applied for thermal boundary layers, using the equation for energy in compressible flows.[7]​[8]​.

The additional term in the turbulent boundary layer equations is known as the Reynolds shear stress and is unknown a priori. Therefore, the solution of the turbulent boundary layer equations requires the use of a "turbulence model", the objective of which is to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of precision and generality of such models is a major obstacle in the successful prediction of turbulent flow properties in modern fluid dynamics.

In the region close to the wall there is a layer of constant tension. Due to the damping of vertical velocity fluctuations near the wall, the Reynolds stress term will be negligible and we will find that a linear velocity profile exists. This is only true for the region very close to the wall.

In 1928, the French engineer André Lévêque") observed that heat transfer by convection in a moving fluid is only affected by velocity values very close to the surface.[9]​[10]​ In flows with a high Prandtl number, the temperature/mass transition from the surface temperature to the free stream temperature takes place in a very thin region close to the surface. Therefore, the most important fluid velocities are those lie within this very fine region in which the change in velocity can be considered linear with the normal distance from the surface. Thus, for.

when, then.

where θ is the tangent of the Poiseuille parabola that intersects the wall.

Although Lévêque's solution was specific to heat transfer in a Poiseuille flow, his knowledge helped other scientists find an exact solution to the thermal boundary layer problem.[11]​ Schuh observed that in a boundary layer, u is again a linear function of y, but that in this case, the wall tangent is a function of x.[12]​ He expressed this with a modified version of Lévêque's profile.

This results in a very good approximation, even for low numbers, so that only liquid metals with very less than 1 cannot be treated in this way.[11]​.

In 1962, Kestin and Persen published a paper in which they described solutions for heat transfer when the thermal boundary layer is completely contained within the momentum layer and for various wall temperature distributions.[13] For the problem of a flat plate with a temperature jump in , they proposed a substitution that reduces the parabolic thermal boundary layer equation to an ordinary differential equation. The solution of this equation, the temperature at any point in the fluid, can be expressed as an incomplete gamma function.[10] Schlichting") proposed an equivalent substitution that reduces the thermal boundary layer equation to an ordinary differential equation whose solution is the same incomplete gamma function.[14]​.

As is well known through various textbooks, heat transfer tends to decrease with increasing boundary layer. Recently, it was observed in a large-scale practical case that wind flowing through a photovoltaic generator tends to "trap" heat in the photovoltaic panels under a turbulent regime due to decreased heat transfer.[15] Although often assumed to be inherently turbulent, this accidental observation demonstrates that natural wind behaves in practice very similarly to an ideal fluid, at least in one observation that resembles the behavior expected on a flat plate, potentially reduces the difficulty of analyzing this type of phenomenon on a larger scale.

Heinrich Blasius derived an exact solution to the above laminar boundary layer equations.[16] The thickness of the boundary layer is a function of the Reynolds number for laminar flow.

The Blasius solution uses dimensionless boundary conditions:

Note that in many cases, the boundary no-slip condition holds that the velocity of the fluid at the surface of the plate is equal to the velocity of the plate everywhere. If the plate does not move, then . A much more complicated bypass is required if fluid slippage is allowed.[17]​.

In fact, Blasius' solution for the laminar velocity profile in the boundary layer over a semi-infinite plate can be easily extended to describe the Thermal and Concentration boundary layers for heat and mass transfer respectively. Instead of the differential x-momentum balance (equation of motion), a similarly derived Energy and Mass balance is used:

Energy:.

Mass:.

For momentum balance, kinematic viscosity can be considered as momentum diffusivity. In the energy balance it is replaced by thermal diffusivity, and by mass diffusivity in the mass balance. In the thermal diffusivity of a substance, it is its thermal conductivity, it is its density and it is its heat capacity. The subscript AB denotes the diffusivity of species A that diffuses into species B.

Under the assumption that , these equations become equivalent to the momentum balance. Thus, for the Prandtl number and the Schmidt number the Blasius solution applies directly.

Consequently, this derivation uses a related form of the boundary conditions, substituting or (absolute temperature or concentration of species A). The subscript S denotes a surface condition.

Using the streamline function Blasius obtained the following solution for the shear stress on the surface of the plate.

And through the boundary conditions, we know that.

The following relations are given for the heat/mass flux off the surface of the plate.

So stop.

where are the flow regions where and are less than 99% of their far-field values.[18]​.

Since the Prandtl number of a particular fluid is usually not unity, the German engineer E. Polhausen, who worked with Ludwig Prandtl, attempted to empirically extend these equations to apply them for . His results can also be applied to .[19]​ He discovered that for a Prandtl number greater than 0.6, the thickness of the thermal boundary layer was approximately given by:.

From this solution, it is possible to characterize the convective heat/mass transfer constants as a function of the boundary layer flow region. Fourier's Conduction Law#Fourier's_Law "Conduction (heat)") and Newton's Law of Cooling are combined with the flow term derived above and the thickness of the boundary layer.

This gives the local convective constant at a point on the semi-infinite plane. Integrating over the length of the plate gives an average.

Following the derivation with mass transfer terms ( = convective mass transfer constant, = diffusivity of species A in species B, ), the following solutions are obtained:.

These solutions are valid for a laminar flow with a Prandtl/Schmidt number greater than 0.6.[18]​.

This effect was exploited in the Tesla turbine, patented by Nikola Tesla in 1913. It is called a bladeless turbine because it uses the boundary layer effect and not a fluid that impinges on the blades as in a conventional turbine. Boundary layer turbines are also known as cohesion turbines, bladeless turbines, and Prandtl layer turbines (after Ludwig Prandtl).

Using the transient and viscous force equations for a cylindrical flow, the thickness of the transient boundary layer can be predicted by finding the Womersley Number ().

Transient force =.

Viscous Force =.

Equating them together we obtain:

Solving for delta gives:

In dimensionless form:.

where = Womersley number; = density; = speed; ?; = length of the transient boundary layer; = viscosity; = characteristic length.

Using the convective and viscous force equations in the boundary layer for a cylindrical flow, the convective flow conditions in the boundary layer can be predicted by finding the dimensionless Reynolds Number ().

Convective force:

Viscous force:

Equating them together we obtain:

Solving for delta gives:

In dimensionless form:.

where = Reynolds number; = density; = speed; = convective boundary layer length; = viscosity; = characteristic length.

La capa límite se estudia para analizar la variación de velocidades en la zona de contacto entre un fluido y un obstáculo que se encuentra en su seno o por el que se desplaza. La presencia de esta capa es debida principalmente a la existencia de la viscosidad, propiedad inherente de cualquier fluido. Esta es la causante de que el obstáculo produzca una variación en el movimiento de las líneas de corriente más próximas a él. El hecho de que la viscosidad sea importante invalida un análisis apresurado en función del principio de Bernoulli del origen de las fuerzas aerodinámicas ya que dicho principio sólo es de aplicación cuando las fuerzas viscosas sean despreciables.

En la atmósfera terrestre, la capa límite es la capa de aire cercana al suelo y que se ve afectada por la convección debida al intercambio diurno de calor, humedad y momento con el suelo.

En el caso de un sólido moviéndose en el interior de un fluido, una capa límite laminar proporciona menor resistencia al movimiento.

In the case of channels

The boundary layer, in hydraulics, is the area of ​​flow in a channel or tube, where the roughness of the tube or channel is strongly felt.

The effect of the boundary layer on the flow can be assimilated to a fictitious upward displacement of the channel bottom to a virtual position. This displacement is called displacement thickness.

At the beginning of the flow in a channel that starts, for example from a reservoir or lake, the flow is entirely laminar. In these situations, a laminar boundary layer develops, the thickness of which increases. From a certain distance from the beginning of the channel, the boundary layer becomes turbulent, without disappearing the laminar boundary layer, whose thickness asymptotically tends to a value that is a function of the speed, the viscosity of the water and the roughness of the walls and bottom of the channel.[20]​.

• - Aerodynamics.

• - Vorticity.

• - Separation of the boundary layer.

• - Boundary layer suction.

• - Blasius boundary layer.

• - Falkner-Skan boundary layer.

• - Perturbational theory.

• - Shear stress.

• - Fernández Díez, Pedro. «VIII.- Elementary theory of the two-dimensional boundary layer», in Fluid Mechanics. Department of Electrical and Energy Engineering. University of Cantabria.

The Aerodynamic boundary layer was first proposed by Ludwig Prandtl in a paper presented on August 12, 1904 at the Third International Congress of Mathematicians in Heidelberg, Germany. It simplifies the equations of fluid flow by dividing the flow field into two zones: one within the boundary layer, dominated by viscosity and creating most of the resistance "Resistance (fluids)") experienced by the boundary body; and another outside the boundary layer, where the viscosity can be neglected without significant effects on the solution. This allows a closed-form (mathematical) closed-form solution for the flow in both zones making significant simplifications of the full Navier-Stokes equations. The same hypothesis is applicable to other fluids (besides air) of moderate to low viscosity, such as water. In the case where there is a temperature difference between the surface and the fluid, it is observed that most of the heat transfer to and from a body takes place in the vicinity of the velocity boundary layer. Again, this allows the equations in the flow field outside the boundary layer to be simplified. The pressure distribution along the boundary layer in the direction normal to the surface (like an airfoil) remains relatively constant along the boundary layer, and is the same as at the surface itself.

The thickness of the velocity boundary layer is normally defined as the distance from the solid body to the point at which the viscous flow velocity is 99% of the free stream velocity (the surface velocity of an inviscid flow). The displacement thickness") is an alternative definition that states that the boundary layer represents a deficit in mass flow compared to non-viscous flow with wall slip. It is the distance the wall would have to be displaced in the non-viscous case to obtain the same total mass flux as in the viscous case. The no-slip condition requires that the flow velocity at the surface of a solid object be zero and that the fluid temperature be equal to the surface temperature. The flow velocity will then increase rapidly within the boundary layer, governed by the boundary layer equations, below.

The thickness of the thermal boundary layer is the distance from the body at which the temperature is 99% of the free stream temperature. The relationship between the two thicknesses is governed by the Prandtl number. If the Prandtl number is 1, the two boundary layers have the same thickness. If the Prandtl number is greater than 1, the thermal boundary layer is thinner than the velocity boundary layer. If the Prandtl number is less than 1, which is the case for air under standard conditions, the thermal boundary layer is thicker than the velocity boundary layer.

In high-performance designs, such as gliders and commercial aircraft, much attention is paid to controlling the behavior of the boundary layer to minimize drag. Two effects must be taken into account. First, the boundary layer increases the effective thickness of the body, through the displacement thickness"), thus increasing the pressure resistance. Second, the shear forces") on the wing surface create skin friction resistance.

At high Reynolds numbers, typical of large aircraft, it is desirable to have a laminar boundary layer. This results in lower skin friction due to the characteristic velocity profile of laminar flow. However, the boundary layer inevitably thickens and becomes less stable as the flow develops along the body, eventually becoming turbulent, a process known as the boundary layer transition. the boundary layer towards the stern by reshaping the airfoil or fuselage so that its thickest point is further aft and less thick. In this way, the speeds at the front are reduced and the same Reynolds number is achieved with a greater length.

At low Reynolds numbers, such as those seen in model airplanes, it is relatively easy to maintain laminar flow. This results in low surface friction, which is desirable. However, the same velocity profile that gives the laminar boundary layer its low skin friction also causes it to be greatly affected by adverse pressure gradients. When pressure begins to recover at the rear of the wing chord, the laminar boundary layer will tend to separate from the surface. Such flow separation causes a large increase in pressure resistance, since it greatly increases the effective size of the wing section. In these cases, it may be advantageous to deliberately cause boundary layer turbulence at a point prior to the laminar separation, using a turbulator. The more complete velocity profile of the turbulent boundary layer allows it to maintain the adverse pressure gradient without separating. Thus, although surface friction increases, overall resistance decreases. This is the principle behind the dimples in golf balls and the vortex generators in airplanes. Special wing sections have also been designed that adapt pressure recovery to reduce or even eliminate laminar separation. This represents an optimal compromise between the pressure resistance of flow separation and the surface friction of induced turbulence.

When using semi-models in wind tunnels, a peniche&action=edit&redlink=1 "Peniche (fluid dynamics) (not yet drafted)") is sometimes used to reduce or eliminate the boundary layer effect.

Contenido

La deducción de las ecuaciones de la capa límite fue uno de los avances más importantes en la dinámica de fluidos. Utilizando un análisis de orden de magnitud"), las conocidas ecuaciones de Navier-Stokes que gobiernan el flujo viscoso de fluidos pueden simplificarse en gran medida dentro de la capa límite. pueden simplificarse en gran medida dentro de la capa límite. En particular, el [característica de la Ecuaciones diferenciales parciales (EDP) se convierte en parabólica, en lugar de la forma elíptica de las ecuaciones de Navier-Stokes completas. Esto simplifica enormemente la solución de las ecuaciones. Al hacer la aproximación de la capa límite, el flujo se divide en una parte no viscosa (que es fácil de resolver por varios métodos) y la capa límite, que se rige por una EDP más fácil de resolver. Las ecuaciones de continuidad y de Navier-Stokes para un flujo incompresible bidimensional constante en coordenadas cartesianas vienen dadas por.

donde y son las componentes de la velocidad, es la densidad, es la presión, y es la viscosidad cinemática del fluido en un punto.

La aproximación establece que, para un número de Reynolds suficientemente alto, el flujo sobre una superficie puede dividirse en una región exterior de flujo no viscoso no afectado por la viscosidad (la mayor parte del flujo), y una región cercana a la superficie donde la viscosidad es importante (la capa límite). Sean y las velocidades de la corriente y transversal (normal a la pared) respectivamente dentro de la capa límite. Usando análisis de escala "Análisis de escala (matemáticas)"), se puede demostrar que las ecuaciones de movimiento anteriores se reducen dentro de la capa límite a ser.

y si el fluido es incompresible (como los líquidos en condiciones normales):.

El análisis del orden de magnitud supone que la escala de longitud en el sentido de la corriente es significativamente mayor que la escala de longitud transversal dentro de la capa límite. De ello se deduce que las variaciones de las propiedades en la dirección de la corriente son generalmente mucho menores que las de la dirección normal de la pared. Si se aplica esto a la ecuación de continuidad, se observa que , la velocidad normal de la pared, es pequeña comparada con , la velocidad en el sentido de la corriente.

Como la presión estática es independiente de , entonces la presión en el borde de la capa límite es la presión en toda la capa límite en una posición dada en el sentido de la corriente. La presión externa puede obtenerse mediante una aplicación de la ecuación de Bernoulli. Sea la velocidad del fluido fuera de la capa límite, donde y son ambas paralelas. Esto da al sustituir por el siguiente resultado.

Para un flujo en el que la presión estática tampoco cambia en la dirección del flujo.

si permanece constante.

Por lo tanto, la ecuación de movimiento se simplifica y pasa a ser.

Estas aproximaciones se utilizan en una variedad de problemas prácticos de flujo de interés científico y de ingeniería. El análisis anterior es para cualquier laminar instantáneo o turbulento, pero se utiliza principalmente en estudios de flujo laminar ya que el flujo medio es también el flujo instantáneo porque no hay fluctuaciones de velocidad presentes. Esta ecuación simplificada es una EDP parabólica y puede resolverse utilizando una solución de similitud a menudo denominada capa límite de Blasius.

Prandtl transposition theorem

Prandtl observed that from any solution that satisfies the boundary layer equations, another solution, which also satisfies the boundary layer equations, can be constructed by writing [2]​.

where it is arbitrary. Since the solution is not unique from a mathematical point of view,[3]​ any of an infinite set of eigenfunctions can be added to the solution as demonstrated by Stewartson").[4]​ and Paul A. Libby").[5]​[6]​.

The treatment of turbulent boundary layers is much more difficult due to the time-dependent variation of flow properties. One of the most used techniques to treat turbulent flows is to apply the Reynolds decomposition. In it, the instantaneous properties of the flow are decomposed into an average and a fluctuating component, assuming that the average of the fluctuating component is always zero. Applying this technique to the boundary layer equations, the complete turbulent boundary layer equations are obtained, which are rare in the literature:

Using a similar order of magnitude analysis, the above equations can be reduced to leading order terms. Choosing length scales for changes in the transverse direction, and for changes in the direction of the current, with , the x-momentum equation simplifies to:.

This equation does not satisfy the no-slip condition on the wall. As Prandtl did for his boundary layer equations, a new smaller length scale must be used to allow the viscous term to become leading order in the momentum equation. Choosing y as the scale, the leading order moment equation for this "inner boundary layer" is given by:.

In the limit of infinite Reynolds number, it can be shown that the pressure gradient term has no effect in the inner region of the turbulent boundary layer. The new "internal length scale" is a viscous length scale, and is of order , being the speed scale of turbulent fluctuations, in this case a frictional speed.

Unlike the laminar boundary layer equations, the presence of two regimes governed by different sets of flow scales (i.e., the inner and outer scale) has made finding a universally similar solution for the turbulent boundary layer difficult and controversial. To find a similarity solution that spans both regions of the flow, it is necessary to asymptotically match the solutions of both regions of the flow. This analysis will result in the so-called Wall Law or Power Law Solutions.

Approaches similar to the previous analysis have also been applied for thermal boundary layers, using the equation for energy in compressible flows.[7]​[8]​.

The additional term in the turbulent boundary layer equations is known as the Reynolds shear stress and is unknown a priori. Therefore, the solution of the turbulent boundary layer equations requires the use of a "turbulence model", the objective of which is to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of precision and generality of such models is a major obstacle in the successful prediction of turbulent flow properties in modern fluid dynamics.

In the region close to the wall there is a layer of constant tension. Due to the damping of vertical velocity fluctuations near the wall, the Reynolds stress term will be negligible and we will find that a linear velocity profile exists. This is only true for the region very close to the wall.

In 1928, the French engineer André Lévêque") observed that heat transfer by convection in a moving fluid is only affected by velocity values very close to the surface.[9]​[10]​ In flows with a high Prandtl number, the temperature/mass transition from the surface temperature to the free stream temperature takes place in a very thin region close to the surface. Therefore, the most important fluid velocities are those lie within this very fine region in which the change in velocity can be considered linear with the normal distance from the surface. Thus, for.

when, then.

where θ is the tangent of the Poiseuille parabola that intersects the wall.

Although Lévêque's solution was specific to heat transfer in a Poiseuille flow, his knowledge helped other scientists find an exact solution to the thermal boundary layer problem.[11]​ Schuh observed that in a boundary layer, u is again a linear function of y, but that in this case, the wall tangent is a function of x.[12]​ He expressed this with a modified version of Lévêque's profile.

This results in a very good approximation, even for low numbers, so that only liquid metals with very less than 1 cannot be treated in this way.[11]​.

In 1962, Kestin and Persen published a paper in which they described solutions for heat transfer when the thermal boundary layer is completely contained within the momentum layer and for various wall temperature distributions.[13] For the problem of a flat plate with a temperature jump in , they proposed a substitution that reduces the parabolic thermal boundary layer equation to an ordinary differential equation. The solution of this equation, the temperature at any point in the fluid, can be expressed as an incomplete gamma function.[10] Schlichting") proposed an equivalent substitution that reduces the thermal boundary layer equation to an ordinary differential equation whose solution is the same incomplete gamma function.[14]​.

As is well known through various textbooks, heat transfer tends to decrease with increasing boundary layer. Recently, it was observed in a large-scale practical case that wind flowing through a photovoltaic generator tends to "trap" heat in the photovoltaic panels under a turbulent regime due to decreased heat transfer.[15] Although often assumed to be inherently turbulent, this accidental observation demonstrates that natural wind behaves in practice very similarly to an ideal fluid, at least in one observation that resembles the behavior expected on a flat plate, potentially reduces the difficulty of analyzing this type of phenomenon on a larger scale.

Heinrich Blasius derived an exact solution to the above laminar boundary layer equations.[16] The thickness of the boundary layer is a function of the Reynolds number for laminar flow.

The Blasius solution uses dimensionless boundary conditions:

Note that in many cases, the boundary no-slip condition holds that the velocity of the fluid at the surface of the plate is equal to the velocity of the plate everywhere. If the plate does not move, then . A much more complicated bypass is required if fluid slippage is allowed.[17]​.

In fact, Blasius' solution for the laminar velocity profile in the boundary layer over a semi-infinite plate can be easily extended to describe the Thermal and Concentration boundary layers for heat and mass transfer respectively. Instead of the differential x-momentum balance (equation of motion), a similarly derived Energy and Mass balance is used:

Energy:.

Mass:.

For momentum balance, kinematic viscosity can be considered as momentum diffusivity. In the energy balance it is replaced by thermal diffusivity, and by mass diffusivity in the mass balance. In the thermal diffusivity of a substance, it is its thermal conductivity, it is its density and it is its heat capacity. The subscript AB denotes the diffusivity of species A that diffuses into species B.

Under the assumption that , these equations become equivalent to the momentum balance. Thus, for the Prandtl number and the Schmidt number the Blasius solution applies directly.

Consequently, this derivation uses a related form of the boundary conditions, substituting or (absolute temperature or concentration of species A). The subscript S denotes a surface condition.

Using the streamline function Blasius obtained the following solution for the shear stress on the surface of the plate.

And through the boundary conditions, we know that.

The following relations are given for the heat/mass flux off the surface of the plate.

So stop.

where are the flow regions where and are less than 99% of their far-field values.[18]​.

Since the Prandtl number of a particular fluid is usually not unity, the German engineer E. Polhausen, who worked with Ludwig Prandtl, attempted to empirically extend these equations to apply them for . His results can also be applied to .[19]​ He discovered that for a Prandtl number greater than 0.6, the thickness of the thermal boundary layer was approximately given by:.

From this solution, it is possible to characterize the convective heat/mass transfer constants as a function of the boundary layer flow region. Fourier's Conduction Law#Fourier's_Law "Conduction (heat)") and Newton's Law of Cooling are combined with the flow term derived above and the thickness of the boundary layer.

This gives the local convective constant at a point on the semi-infinite plane. Integrating over the length of the plate gives an average.

Following the derivation with mass transfer terms ( = convective mass transfer constant, = diffusivity of species A in species B, ), the following solutions are obtained:.

These solutions are valid for a laminar flow with a Prandtl/Schmidt number greater than 0.6.[18]​.

This effect was exploited in the Tesla turbine, patented by Nikola Tesla in 1913. It is called a bladeless turbine because it uses the boundary layer effect and not a fluid that impinges on the blades as in a conventional turbine. Boundary layer turbines are also known as cohesion turbines, bladeless turbines, and Prandtl layer turbines (after Ludwig Prandtl).

Using the transient and viscous force equations for a cylindrical flow, the thickness of the transient boundary layer can be predicted by finding the Womersley Number ().

Transient force =.

Viscous Force =.

Equating them together we obtain:

Solving for delta gives:

In dimensionless form:.

where = Womersley number; = density; = speed; ?; = length of the transient boundary layer; = viscosity; = characteristic length.

Using the convective and viscous force equations in the boundary layer for a cylindrical flow, the convective flow conditions in the boundary layer can be predicted by finding the dimensionless Reynolds Number ().

Convective force:

Viscous force:

Equating them together we obtain:

Solving for delta gives:

In dimensionless form:.

where = Reynolds number; = density; = speed; = convective boundary layer length; = viscosity; = characteristic length.

La capa límite se estudia para analizar la variación de velocidades en la zona de contacto entre un fluido y un obstáculo que se encuentra en su seno o por el que se desplaza. La presencia de esta capa es debida principalmente a la existencia de la viscosidad, propiedad inherente de cualquier fluido. Esta es la causante de que el obstáculo produzca una variación en el movimiento de las líneas de corriente más próximas a él. El hecho de que la viscosidad sea importante invalida un análisis apresurado en función del principio de Bernoulli del origen de las fuerzas aerodinámicas ya que dicho principio sólo es de aplicación cuando las fuerzas viscosas sean despreciables.

En la atmósfera terrestre, la capa límite es la capa de aire cercana al suelo y que se ve afectada por la convección debida al intercambio diurno de calor, humedad y momento con el suelo.

En el caso de un sólido moviéndose en el interior de un fluido, una capa límite laminar proporciona menor resistencia al movimiento.

In the case of channels

The boundary layer, in hydraulics, is the area of ​​flow in a channel or tube, where the roughness of the tube or channel is strongly felt.

The effect of the boundary layer on the flow can be assimilated to a fictitious upward displacement of the channel bottom to a virtual position. This displacement is called displacement thickness.

At the beginning of the flow in a channel that starts, for example from a reservoir or lake, the flow is entirely laminar. In these situations, a laminar boundary layer develops, the thickness of which increases. From a certain distance from the beginning of the channel, the boundary layer becomes turbulent, without disappearing the laminar boundary layer, whose thickness asymptotically tends to a value that is a function of the speed, the viscosity of the water and the roughness of the walls and bottom of the channel.[20]​.

• - Aerodynamics.

• - Vorticity.

• - Separation of the boundary layer.

• - Boundary layer suction.

• - Blasius boundary layer.

• - Falkner-Skan boundary layer.

• - Perturbational theory.

• - Shear stress.

• - Fernández Díez, Pedro. «VIII.- Elementary theory of the two-dimensional boundary layer», in Fluid Mechanics. Department of Electrical and Energy Engineering. University of Cantabria.